There is a massive difference between an emotion, and the idea of that same emotion. The problem is that people who are bad with emotions are also bad at perceiving the distinction between the two. This is incredibly clear with fear, as a lot of emotions can be entirely downstream of fear, such as:
Anger
Blame
Outrage
Sadness, depression, despair
Hope
Guilt or shame
Impatience, restlessness
Neediness, addiction
Control
I would say there are several reasons why the fear hides itself behind other emotions. The first one is that basically no one wants to admit that they're afraid. Admitting that you cannot talk to girls, don't know how to draw or dance, are addicted to porn and video games, do not trust your partner, are very emotionally brittle or that you are always restless, those can all be very embarrassing, but still, people sometimes confess about those things. On the other hand, admitting that any of those things is downstream of fear: the fear of rejection, the fear of other people's opinions, or even the fear of living, feeling and having free time, those are far more confronting. People would rather be losers, incompetent, selfish or even insane, than be cowards, because it feels so personal. At least when you're incompetent or selfish or a loser, you could argue that certain things are out of your control, and that you were dealt a bad hand. But when you're afraid of taking action? There's a visceral sense that you are weak.
The second, more existential reason is that the fear has a sort of intelligence, that can slip underneath the conscious layer of the mind. Ego is constantly looking for danger, and thus constantly afraid, because it knows ultimately that it is built on a shaky foundation. Not just the fact that it will die, which is of course true and something that very few people actually acknowledge in their guts (not just their mind) but the existential fear of the ego is broader than that. It is firstly afraid of any deep change: challenging work, taking risks at another path in life, breaking out of old habits, removing unhealthy relationships from your life, etc. But it is also deeply afraid of any deep examination, because it is pretty damn selfish. Acquiring things—which includes knowledge, when used as an armor to feel superior to others for instance—controlling people and events, only paying attention to others when it benefits you, etc.9
We could summarize as such: ego is a fragile shell, but one that has a sort of intelligence—your intelligence in fact—that allows it to maintain itself in a more active manner. Ultimately that shell will die out, and it is also intelligent enough to know that, but will never look at that truth, and many other small truths, dead in the face. This is why egoic people distract themselves all the fucking time. Distraction is a fantastic mechanism for the ego because it feels like it is doing something useful, when for instance talking about politics, getting better at a video game, building an intellectual armor out of meaningless facts, etc. But ultimately that distraction is born out of fear, which is why people are never fully at peace when they engage in modern entertainment. Deep down, they know they are bullshitting themselves, and running away from something important.